The existence of the evolution system \(\lbrace  E_m (t,s) \rbrace \)  will directly follow from Lemma REF  once we prove that \(A_m(t)\)  satisfies property (S).
Fix \(t\in [0,T].\)  It follows from [1]} that the operator \(A_m(t) = \mathbb {P}_{m}A(t) \mathbb {P}_{m}+ (I-\mathbb {P}_{m}) B (I-\mathbb {P}_{m})\)  generates a cosine function on \(H,\)  since \(\mathbb {P}_{m}A(t) \mathbb {P}_{m}\)  is bounded and \((I-\mathbb {P}_{m})B(I-\mathbb {P}_{m})\)  is symmetric.
By [2]}, the numerical range \(W(A_m(t)):=\lbrace  \langle A_m(t) u, u \rangle | u\in V, \Vert u \Vert _H=1\rbrace \)  is then contained in a parabola.
Lastly, by [3]} it follows that the square root condition \(D(A_m(t)^{1/2} )=V\)  holds (see also [4]}). This proves the first part.
Let \(x\in V_m\)  and let \(u(t)\)  be the solution of (REF ). Note that the function \(t\rightarrow u(t)-\mathbb {P}_{m}u(t)\)  has a.e. derivative \( u^{\prime }(t) -\mathbb {P}_{m}u^{\prime }(t)\)  since \(\mathbb {P}_{m}\)  is linear; moreover it holds
\(\nonumber && \frac{1}{2} \frac{d}{dt} \Vert  u(t) - \mathbb {P}_{m}u(t) \Vert _H^2 = \Re \langle u^{\prime } (t) - \mathbb {P}_{m}u^{\prime } (t), u(t)- \mathbb {P}_{m}u(t) \rangle \\\nonumber && = - \Re \langle A_m (t) u(t), u(t)-\mathbb {P}_{m}u(t) \rangle + \Re \langle \mathbb {P}_{m}A_m (t) u(t), u(t)-\mathbb {P}_{m}u(t) \rangle \\\nonumber && = - \Re \langle A_m (t) u(t), u(t)-\mathbb {P}_{m}u(t) \rangle + \Re \langle A_m (t) u(t), \mathbb {P}_{m}( u(t)-\mathbb {P}_{m}u(t) ) \rangle \\\nonumber && = -\Re \langle A_m (t) u(t), u(t)-\mathbb {P}_{m}u(t) \rangle \\\nonumber && = - a_m (t, u (t), u(t) - \mathbb {P}_{m}u (t) )\\&& = -a (t, \mathbb {P}_{m}u(t), \mathbb {P}_{m}( I-\mathbb {P}_{m}) u(t)) - \alpha \Vert  (I-\mathbb {P}_{m}) u(t) \Vert _V^2,\) 
where we have used the fact that \(\mathbb {P}_{m}\)  is self-adjoint w.r.t \(\langle \cdot , \cdot \rangle \)  and \(I-\mathbb {P}_{m}\)  is idempotent. Note that \(\mathbb {P}_{m}(I-\mathbb {P}_{m}) = 0,\)  which implies that \(a (t, \mathbb {P}_{m}u(t), \mathbb {P}_{m}( I-\mathbb {P}_{m}) u(t))=0\)  and (REF ) brings to
\(\frac{d}{dt} \Vert  u(t) - \mathbb {P}_{m}u(t) \Vert _H^2 \le 0 - \alpha \Vert  (I-\mathbb {P}_{m}) u(t) \Vert _V^2 \le 0.\) 
The direct consequence of this last is that for a.e. \(t\in [0,T],\)  \( \Vert  u(t) - \mathbb {P}_{m}u(t)\Vert _H \le \Vert  x - \mathbb {P}_{m}x\Vert _H=0\)  since \(x\in V_m.\)  Thus \(u(t)=\mathbb {P}_{m}u(t)\)  and \(u(t)\in V_m\)  as a consequence.
